Ladzoryshyn N. Equivalence of matrices over quadratic rings and matrix equations.

In the thesis it is investigated the special equivalence of matrices and their pairs over quadratic rings. We call it (z,k)-equivalence. This triangular matrix TA is called the standard form by the matrix A with respect to the (z,k)-equivalence. It is shown that the number of standard forms of the matrix over the Euclidean imaginary quadratic rings is finite. It is found that not every pair of matrices over a quadratic ring is (z,k)-equivalent to a pair of matrices in standard forms. It is proved, that a pair of matrices over the Euclidean quadratic rings and the quadratic principal ideal rings with determinants are relatively prime or are degrees simple numbers in a quadratic ring can be reduced by means of (z,k)-equivalent transformations to standard forms. The conditions for solvability of matrix unilateral and bilateral equations over quadratic rings and a method of finding solutions to these equations is given. Integer solutions are described, that is, the solutions with elements of the ring of integers of the matrix equations. The criteria for the existence of integer solutions and their uniqueness are given. The established standard forms of matrices and their pairs are applied for construction of effective methods for solving these matrix equations and study of the structure of their solutions. It is shown that solvable matrix equations have solutions with limited of Euclidean norms and such solutions of the matrix equation over the Euclidean imaginary quadratic rings are the finite number.